Assuming Complex Shape a Simple Shape. What is this called?

In summary, topology is the study of continuity and deformations in a mathematical context. It allows us to describe shapes and objects without relying on specific geometric properties, such as distance or angles. Simplifying assumptions, like assuming a complex shape to be a simpler one, are not well-defined mathematical notions but can still involve interesting mathematical questions. This is not considered topology, as it does not involve the same concepts and properties that topology focuses on.
  • #1
chiako
11
0
What topic in mathematics would something fall under if you assumed a complex shape to be a more simple shape? Such as assuming a recliner is actually a cube, or a tree it actually a cone or prism, or a tire is actually a cylinder? Would this be topology, or something else? Simple geometry?
 
Physics news on Phys.org
  • #2
Hey chiako and welcome to the forums.

This is essentially what topology does. Topology allows you to describe continuity in a way that deals with deformations of an object or representation.

So yes when you have a situation where you deform something but can't change the topology you can't say turn a donut into a ball because that would mean getting rid of the hole. This isn't really a rigorous explanation but I think you get the idea.

Again with something like contuinity, if you deform a continuous thing without change its topology it should stay continuous. There are many ways of describing this concept in a variety of contexts.

Also there are different kinds of topologies applied to a variety of fields which have their own ideas and nonclamenture.
 
  • #3
I would call these "simplifying assumptions" rather than any kind of well-defined mathematical notion, since it seems like you are wanting the approximating shapes to have some geometric relationship to the things they approximate. There are lots of different mathematical notions related to "shape" (or, more typically, we think about different "spaces"), and they're characterized by the set of properties we choose to care about. For example, metric spaces care about distances, topological spaces care about continuity, Euclidean space cares about angles and distances (what non-math/physics folks have in mind when they think of "geometry"), projective space has its own set of invariant quantities that aren't so intuitive, and so on.

So in a topological space, we "don't care about distance", or more accurately, distance just doesn't exist there. The usual example is that you can turn a coffee mug into a donut, but it's more precise to say that they were already the same thing to begin with--the things that let us distinguish coffee mugs from donuts have to do with thickness, angles, concavity, and so forth, and these concepts just don't have any definition in a topological space. But if I take that donut and tear it into two pieces, then I can tell that I did something, because topological space does care about connectedness. (I can also tell that the pieces themselves are not donuts, using some other topological tools.)

So, back to your particular case: It seems like you are essentially "caring about" all the same things for your approximating objects (sizes, angles, overall shape) as you do for the original objects. So you aren't looking at them in a different context mathematically, but rather just "simplifying" them. Of course, doing this in an automatic, efficient, consistent way probably raises all sorts of interesting mathematical questions, so I'm not saying math isn't involved. Probably not topology, though.
 

1. What is meant by "assuming complex shape a simple shape"?

Assuming complex shape a simple shape refers to the process of simplifying or approximating a complex shape by breaking it down into smaller, simpler shapes that are easier to analyze or work with.

2. Why would someone want to assume complex shape a simple shape?

Assuming complex shape a simple shape is often done for practical reasons, such as simplifying calculations or creating more efficient designs. It can also help in understanding the underlying structure or patterns of a complex shape.

3. What are some common methods used to assume complex shape a simple shape?

Some common methods include using geometric approximations, such as circles or rectangles, to represent more complex shapes. Other methods include using mathematical equations or computer algorithms to simplify the shape.

4. Is assuming complex shape a simple shape always accurate?

No, assuming complex shape a simple shape is not always accurate. It is a simplification technique and may not capture all the nuances or details of the original complex shape. It is important to evaluate the level of accuracy needed for the specific application.

5. Can assuming complex shape a simple shape be applied in all scientific fields?

Yes, the concept of assuming complex shape a simple shape can be applied in various scientific fields, such as physics, chemistry, biology, and engineering. It is a fundamental concept in mathematical modeling and data analysis.

Similar threads

  • Electrical Engineering
Replies
6
Views
1K
Replies
16
Views
471
Replies
1
Views
602
  • Electromagnetism
Replies
3
Views
1K
Replies
13
Views
878
Replies
4
Views
6K
Replies
21
Views
5K
Replies
12
Views
2K
  • Sci-Fi Writing and World Building
Replies
10
Views
2K
  • Differential Equations
Replies
1
Views
1K
Back
Top